29 votes
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Variance replication using options

Let $t_0, t_1, \ldots, t_n$ be observation dates, where $0=t_0 < \cdots < t_n = T$, and $\{S_t \mid t \geq 0\}$ be the equity price process without dividend payments. Then the realized variance ...
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  • 20.4k
23 votes

Value of Call Option as Volatility goes to Infinity

The value of a call option does not go to infinity as the volatility goes to infinity. It tends to the discounted value of the forward $F=S_0 e^{(r-q)T}$, which when the dividend yield is zero, ...
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  • 2,069
19 votes
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Why parameterize the Black Scholes implied volatility surface?

There is no "plain Black Scholes implied surface" because implied volatilities come from options market prices (calls and put). If you had a whole continuum of call prices $C : \mathbb{R}_+ \times \...
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  • 3,826
18 votes
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Arbitrage Free Volatility Smile

I generally agree with @dm63's answer: A convex (concave) smile around the forward usually indicates and leptokurtic (platykurtic) implied risk-neutral probability density. Both situations can or ...
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17 votes
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Is there anywhere I can read the paper, "The Gamma-Vanna-Volga Cost Framework for Constructing Implied Volatility Curves"

Partly because it's hard to get a hold of, the Arslan et. al. paper is starting to assume mythical proportions. As said by Dimitri Vulis, the general idea of the paper is set out in (one or two of) ...
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16 votes
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Volatility arbitrage - how is the profit extracted?

Setting aside, that it's not pure riskless arbitrage, but rather statistical arbitrage: You can extract the profit by performing continuous delta hedging. If you constantly adjust your hedge position ...
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  • 683
16 votes
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Local vol, stochastic vol, implied vol

Along with Gatheral's book, I'd recommend reading Lorenzo Bergomi's "Stochastic Volatility Modelling". The first 2 chapters are available for download on his website. That being said, let me try to ...
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16 votes
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Gamma Pnl vs Vega Pnl

For an option with price $C$, the P$\&$L, with respect to changes of the underlying asset price $S$ and volatility $\sigma$, is given by \begin{align*} P\&L = \delta \Delta S + \frac{1}{2}\...
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15 votes
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Is volatility for the next day forecastable? To any extent?

Upon close reading, this appears to be 3 (interesting) questions, not one. I'm not sure if the mods have the tools needed to split it up, so I'm just going to write down the three questions as I see ...
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15 votes
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What is the connection between the risk neutral implied density and the real world density?

I'll outline how you can estimate the (implied) real-world density function from (observed) option prices. Having found this real-world density, you can then compute all sorts of probabilities and ...
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14 votes
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Why an option has sometimes and implied volatility greater than 100%?

It seems that you are thinking of the volatility as some sort of standard deviation of your stock price. It is not. In the BS model, $\sigma\sqrt{T}$ is the standard deviation of the log-return $\...
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14 votes
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Estimate Beta of CAPM from Implied Volatility?

Yes it is a better way. Just take a look to figure 3, from Buss and Vilkov (2012, RFS):
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  • 6,850
13 votes
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Option Price vs. Implied Volatility

I think it's interesting to look at this problem graphically also. I get a different answer, depending on whether the option is ITM, ATM, or OTM. In the plot below, all options have 1-year expiry, ...
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  • 5,106
12 votes
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Why linear interpolation not appropriate for volatility surface construction?

Note that total implied variance defined as $$ V(T,K) = T\Sigma(T,K)^2 $$ should be an increasing function of $T$. Otherwise you have a calendar arbitrage (sell the call with shorter expiry and ...
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12 votes
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SSR definition in Bergomi in relation to sticky strike and sticky delta

Some Notations It's easy to get lost so let's introduce some notations and let $$ \sigma : (t, S, K, \tau) \to \sigma(K,\tau; S, t) $$ denote the implied volatility smile prevailing at time $t$ ...
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12 votes
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Bergomi: Skew arbitrage

Great question. Let me try to provide some insights and thoughts regarding the points and questions you raised. It may not be a full answer but hopefully it will help connecting the contents in the ...
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  • 721
11 votes
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How to exploit calendar arbitrage?

The answer by @HenriK is certainly correct. However, for justification, technique such as the Jensen inequality is needed. For example, since $x^+$ is a convex function, assuming zero interest and ...
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11 votes
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Why is realized volatility typically lower than implied volatility?

Consider what happens when IV is lower than realised vol. The person long the IV would make money. So there would ideally be no one selling IV if it's lower than realised vol on an average. Next if ...
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11 votes

Forward implied volatility

From an equities perspective, there are two concepts that should not be confused in my opinion and context should make the distinction self-explicit: Forward variance swap volatility (A) Forward ...
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  • 13.8k
10 votes
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For pricing, what types of Exotic Options are suitable using Local Volatility Model or a Stochastic Volatility Model?

Whenever you use any model to price anything, all you need to do is make sure you model the underlying dynamics that the product you're pricing actually depends on. Any product will be dependent on ...
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  • 2,406
10 votes

What is a regime switch?

Regime switching is another way to describe structural changes in a data series. For example, an inflation timeseries may change states from ARMA to linear as the economy moves from a period of ...
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  • 929
10 votes

Why is there greater demand for OTM and ITM options than for ATM options?

Either you or some reference you are following is in error here. At-the-money (or at least near-the-money) options are the most liquidly traded. And trading is much more heavy in out-of-the-money ...
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  • 1,865
10 votes

Implied Volatility of stock on Think or Swim

What they gave you is Newton's formula. If you have a function $f(x)$ then you can find the value $x_0$ such that $f(x_0) = 0$ by this method. It uses the derivative $f'$ which in your case is the ...
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  • 13.3k
9 votes
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Why a calendar spread is a preferred strategy in a low volatility period

The main thing to keep in mind with all these different option combination strategies is that you are really trading option greeks! I think the answer to why the calender spread is so popular lies in ...
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  • 26.9k
9 votes
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How to approximate the time to mean reversion for implied volatility

A very popular choice for mean reversion is the Ornstein–Uhlenbeck process (here in discretized form): $$L_{t+1}-L_t=\alpha(L^*-L_t)+\sigma\epsilon_t$$ Here you see that the level change is governed ...
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9 votes
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Delta Hedging with fixed Implied Volatility or floating Implied Volatility?

Generally speaking, in the real world, you'd always want to use the correct implied vol. But you should think of your question in terms of: (1) Vega mark-to-market (m2m) PnL vs. theta/gamma profile (...
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  • 705
9 votes
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What is implied volatility?

In practice, an implied volatility always refers to the volatility that you need to plug into the Black-Scholes', or Black's, pricing formula to obtain the market price. You may have a different model ...
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  • 20.4k
9 votes
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What is a regime switch?

The idea of regime switching in volatility is rooted in the observation that volatility is usually fairly consistent and "mild", and occasionally very high, say during a market crash. The concept goes ...
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  • 288
9 votes

Why is realized volatility typically lower than implied volatility?

Another theory is that stock indexes do not follow a the lognormal distribution assumption of Black-Scholes. For example U.S. equities measured by the Russell 3000 or S&P 500 have a negative skew ...
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9 votes
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Options Market Making Used Implied Volatility Surface

Your question is twofold How a market maker should adjust its quotes on a vol surface with respect to his inventory? How to adjust the vol surface when a new trade is observed on the markets? Let me ...
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